Czech Technical University inPrague F3 ... · Department of Cybernetics . ... The platform will be used for educational and research experiments ... 1 Introduction 1

Bachelor Project

CzechTechnicalUniversityin Prague

F3 Faculty of Electrical EngineeringDepartment of Cybernetics

Rijke’s Tube – An Experimental Platformfor Modeling and Control inThermoacoustics

Lukáš Černý

Supervisor: doc. Ing. Zdeněk Hurák Ph.D.Field of study: Cybernetics and RoboticsSubfield: RoboticsMay 2017

ii

Czech Technical University in Prague Faculty of Electrical Engineering

Department of Cybernetics

BACHELOR PROJECT ASSIGNMENT

Student: Lukáš Č e r n ý

Study programme: Cybernetics and Robotics

Specialisation: Robotics

Title of Bachelor Project: Rijke's Tube - An Experimental Platform for Modeling and Control in Thermoacoustics

Guidelines: The ultimate task is to build a laboratory experimental platform known as Rijke's tube, which is used for experiments in thermoacoustics. An inspiration can be taken from a description of one particular setup in [1]. The platform will be used for educational and research experiments in modeling, analysis and control of spatially distributed systems. The work will consist of selectrion of suitable components (glass tube, heat source, microphone as an acoustic pressure sensor, loudspeaker as an actuator, flow velocity sensor and perhaps some more), their assembly, design and realization of the electronics for data acquisition, and finally design and implementation of a feedback regulation loop. Finally, some experiments will be conducted and documented to demonstrate the functionality of the platform. Bibliography/Sources: [1] Epperlein, J.P., B. Bamieh, and K.J. Astrom: Thermoacoustics and the Rijke Tube: Experiments, Identification, and Modeling. IEEE Control Systems 35, no. 2 (April 2015): 57-77. doi:10.1109/MCS.2014.2384971. [2] Whitham, G. B.: Linear and Nonlinear Waves. 1 edition. New York, NY: Wiley-Interscience, 1999.

Bachelor Project Supervisor: doc. Ing. Zdeněk Hurák, Ph.D.

Valid until: the end of the summer semester of academic year 2017/2018

L.S.

prof. Dr. Ing. Jan Kybic Head of Department

prof. Ing. Pavel Ripka, CSc. Dean

Prague, January 26, 2017

iv

Acknowledgements

I would like to honestly thank my supervi-sor for his patience, support, and valuableadvice that helped me greatly during mywork on this thesis.

Declaration

I declare that the presented work was de-veloped independently and that I havelisted all sources of information usedwithin it in accordance with the methodi-cal instructions for observing the ethicalprinciples in the preparation of universitytheses.

Prague, date . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .signature

v

Abstract

This work outlines one particular designand assembly of Rijke’s tube with intro-duced feedback in the form of a micro-phone and a loudspeaker. The feedbackwas used to control and stabilize the sys-tem; thus, loud humming that is typi-cal for Rijke’s tube was completely sup-pressed. Further, a few experiments aredocumented to show the basic proper-ties of Rijke’s tube. These are only areconstruction of experiments presentedin available literature.

Keywords: Rijke’s tube,thermoacoustics, acoustics, experimentalplatform, dynamic system, feedbackcontrol, heating coil

Supervisor: doc. Ing. Zdeněk HurákPh.D.Department of Control Engineering,Karlovo náměstí 13/E,Praha 2

Abstrakt

Tato práce popisuje návrh a sestaveníRijkeho trubice včetně zavedení zpětnévazby v podobě mikrofonu a reproduktoru.Takto zavedená zpětná vazba byla použitak řízení a stabilizaci systému, což mělo zanásledek úplné potlačení hlasitého hučení,které je pro Rijkeho trubici typické. Dálejsou na několika experimentech ukázányzákladní vlastnosti Rijkeho trubice. Tytojsou pouze rekonstrukcí experimentů po-psaných v dostupné literatuře.

Klíčová slova: Rijkeho trubice,termoakustika, akustika, experimentálníplatforma, dynamický systém,zpětnovazební řízení, topná spirála

Překlad názvu: Rijkeho trubice –Experimentální zařízení pro modelovánía řízení v termoakustice

vi

Contents

1 Introduction 1

1.1 Related Work . . . . . . . . . . . . . . . . . 2

1.2 Mechanism of Rijke’s Tube . . . . . . 2

1.3 Feedback Control . . . . . . . . . . . . . . 3

2 Construction of Rijke’s Tube 5

2.1 Quartz Glass Tube . . . . . . . . . . . . . 6

2.2 Heating element . . . . . . . . . . . . . . . 6

2.3 Microphone . . . . . . . . . . . . . . . . . . . 7

2.4 Preamplifier . . . . . . . . . . . . . . . . . . . 8

2.5 Speaker and Amplifier . . . . . . . . . 10

2.6 Data Acquisition . . . . . . . . . . . . . 10

3 Experimental Evaluation 13

3.1 Instability Observation . . . . . . . . 13

3.2 Feedback Control . . . . . . . . . . . . . 15

3.3 System Identification . . . . . . . . . . 17

3.4 Heating Coil Temperature . . . . . 18

4 Conclusion 21

A Bibliography 23

B Abbreviations 25

C CD Contents 27

vii

Figures

1.1 An illustration of Rijke’s tube . . . 2

1.2 Fundamental acoustic mode inRijke’s tube (p is pressure, v isacoustic velocity) . . . . . . . . . . . . . . . . 3

1.3 Feedback loop . . . . . . . . . . . . . . . . . 3

2.1 The constructed apparatus . . . . . . 5

2.2 Schematic diagram of the designedpreamplifier . . . . . . . . . . . . . . . . . . . . . 8

2.3 Designed preamplifier assembledon prototyping shield . . . . . . . . . . . . . 8

2.4 Simulink model implementing dataacquisition . . . . . . . . . . . . . . . . . . . . . 11

2.5 Simulink model implementingfeedback loop . . . . . . . . . . . . . . . . . . 11

3.1 Initial growth of oscillations . . . . 14

3.2 Close look at the oscillations . . . 14

3.3 Single-sided amplitude spectrum ofmeasured signal for x = L/4 . . . . . 14

3.4 Single-sided amplitude spectrum ofmeasured signal for x = L/3 . . . . . 15

3.5 Stabilization of the system . . . . . 16

3.6 Higher harmonic instability . . . . 16

3.7 Single-sided amplitude spectrum ofhigher harmonic instability . . . . . . . 17

3.8 Comparision of responses to testsignal . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.9 Comparision of responses to testsignal (close look) . . . . . . . . . . . . . . . 18

3.10 Frequency response from resultsof ARX . . . . . . . . . . . . . . . . . . . . . . . 18

viii

Tables

2.1 Electret microphone MCE-100parameters . . . . . . . . . . . . . . . . . . . . . . 7

2.2 List of components used toassemble the preamplifier . . . . . . . . . 9

2.3 Loudspeaker Monacor SP-10/4parameters . . . . . . . . . . . . . . . . . . . . . 10

ix

Chapter 1

Introduction

Rijke’s tube, or the Rijke tube, is a vertically oriented tube open at bothends with a source of heat energy placed in the lower half. The transferof heat energy from the source can, under the right circumstances, causethermoacoustic instability. This is manifested as oscillations in pressure andvelocity of the gas filling the tube (typically air) which is observed as loudhumming. An important factor is the amount of heat energy transferred fromthe source to the gas. To initiate the instability, the source has to be ableto provide enough energy. In this work, a heating coil connected to voltagesource was used as the heating element. The power consumption of the sourcewas around 380 W.

Rijke’s tube is suitable to be used as an experimental platform for educa-tional purposes. The coupling between heat transfer and acoustics can besubject to mathematical modeling of spatially distributed systems. Havingplaced a microphone at one end of the tube, one can easily investigate thehumming resulting from the unstable coupling. Further, placing a loudspeakerat the other end makes it possible to add an external signal into the system.Moreover, the microphone as a sensor and the loudspeaker as an actuatormay form a feedback loop that can even stabilize the system; thus, suppressthe humming. The feedback loop can also serve to system analysis usingclosed-loop identification techniques.

Description of the construction and particular setup is given in chapter 2.Chapter 3 evaluates some of the performed experiments. It also documentssystem stabilization and identification. Finally, conclusion, final remarks, andpossible directions for future research are mentioned in chapter 4.

1

1. Introduction .....................................1.1 Related Work

Rijke’s tube was discovered by P. L. Rijke in 1859. As he states in [8], heused glass tube with a disc of wire-gauze inside. He used a burner to heatthe disc up. After removing the burner, the tube started to hum loudly. Thehumming went away as the disc was cooling down. Author of [1] and [2] usedelectrically driven heating coil and feedback control techniques to stabilizethe system. In fact, [1] was main inspiration when working on this project.Worth mentioning are also [7] and [5] where horizontally mounted tubes withblowers were used.

1.2 Mechanism of Rijke’s Tube

The presence of the source of heat energy creates an upward flow in the tube(in fig. 1.1 depicted as blue arrow). This flow increases as the heating elementheats up. If the flow is fast enough, oscillations of acoustic velocity andpressure are excited. Usually, fundamental frequency is dominant in theseoscillations. Fundamental frequency is given by the length of of the tube Land the speed of sound in the tube c (approximately 343 m s−1) as

f0 = c

2L. (1.1)

Actually, the oscillations form standing half-wave inside the tube as depictedin fig. 1.2.

Figure 1.1: An illustration of Rijke’s tube

2

...................................1.3. Feedback Control

Figure 1.2: Fundamental acoustic mode in Rijke’s tube (p is pressure, v isacoustic velocity)

1.3 Feedback Control

Feedback loop consisting of a microphone and a loudspeaker (fig. 1.3) enablesthe system analysis and stabilization. The external signal w is used for systemidentification.

Figure 1.3: Feedback loop

3

4

Chapter 2

Construction of Rijke’s Tube

This chapter describes selected components and the process of construction ofRijke’s tube with feedback loop formed of an electret microphone, preamplifier,PC with data acquisition board, power amplifier, and a loudspeaker. Thepreamplifier is one of custom design built on a prototyping shield. Coupleof off-the-shelf electret microphone modules with built-in preamplifiers weretried but all of them happened to either offset or distort measured signal.For this reason, custom preamplifier was designed which turned out to be acheap and effective solution. The final setup of the apparatus is in fig. 2.1.

Figure 2.1: The constructed apparatus

5

2. Construction of Rijke’s Tube ..............................2.1 Quartz Glass Tube

Assuming the heating element to reach high temperatures, quartz glass wasselected as appropriate material for the tube. Quartz glass is fused quartz (orfused silica) without any additives that would lower the melt temperature. Itshigh melting point and low coefficient of thermal expansion make it resistantto thermal shocks. Actually, the tube did not break even if it came in contactwith the heating element. The dimensions of the tube are

L = 1310 mm, (2.1)d = 85 mm (2.2)

where L is its length and d is inner diameter.

To keep the tube in vertical position, custom stand was built. It is madeof aluminium profiles and tube bushings that allow the tube to be fixed indifferent heights. This makes it easy to set a particular distance between thetube and the loudspeaker. Another bushings are used to hold the microphoneand the heating element.

2.2 Heating element

Kanthal wire with diameter 0.64 mm rolled into a coil was used as the heatingelement. The length of the wire was approximately 1.4 m making its resistanceat room temperature

R0 = 6.09 Ω. (2.3)

The coil was connected to voltage source Mean Well SPV-1500-48 (see [6])with output voltage U = 48 V. The maximum current that can be drawnfrom the source is Imax = 32 A. Therefore, its maximum power consumptionis more than 1.5 kW which sufficient to heat up the coil. In fact, the actualpower consumption when the coil is not heated up yet can be calculated as

P = U2

R0= 378.3 W. (2.4)

After heating the coil up, its resistance increases a little (see section 3.4)lowering the power consumption to 373 W.

6

..................................... 2.3. Microphone

It is possible to make the the power consumption higher by making thewire shorter. Nonetheless, a thicker wire should be used because the one usedhere burned at around 500 W. (According to [4], the wire can operate attemperatures up to 1400 C.)

2.3 Microphone

To measure sound signal at the top of the tube, an electret microphoneMCE-100 with an internal FET amplifier was used. This microphone worksas a capacitor that requires connecting to bias voltage. The sound signal isthen measured as voltage changes around the bias voltage. Parameters of themicrophone are in tab. 2.1.

The frequency range of the microphone is sufficient, since higher frequenciesare not expected to be present in the spectrum of measured sound signal.The sensitivity is given by the manufacturer for frequency of 1 kHz only. Forsimplicity’s sake, we can assume the frequency response to be flat through-out the frequency range. However, it would be preferable to measure themicrophone frequency response so we could determine signal distortion.

The microphone was connected to the preamplifier via coaxial cable oflength 1.5 m with audio jack connector at both ends.

Parameter Value

Frequency range 50–10 000 HzSensitivity 5 mV Pa−1 at 1 kHzImpedance 6 kΩ

Operating voltage 1.5–10 VDimensions 9.7 mm × 6.7 mm

Table 2.1: Electret microphone MCE-100 parameters

7

2. Construction of Rijke’s Tube ..............................2.4 Preamplifier

To make the microphone signal measurable by the data acquisition board,it needs to be amplified first. For this purpose, an audio preamplifier wasdesigned. Its schematic is in fig. 2.2 and list of used components is in tab.2.2. When designing the preamplifier, an inspiration was taken from [3]. Fig.2.3 captures the preamplifier assembled on prototyping shield.

Jumper J2 is to provide 12 V power supply for the loudspeaker amplifier.Hence, the amplifier and preamplifier can be powered from a single powersupply.

5

5

4

4

3

3

2

2

1

1

D D

C C

B B

A A

Title

Size Document Number Rev

Date: Sheet of

1

Electret Microphone Preamplifier

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1 1Tuesday, May 16, 2017

Title

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1

Electret Microphone Preamplifier

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1 1Tuesday, May 16, 2017

Title

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Electret Microphone Preamplifier

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R547k

+

C12

10u

R447k

C1100n

R310k

-

+

U2

OP07

3

26

74 8

1

+

C101u

C3100n

C11

100n

C6100n

C7100n

J112V Input

12

+

C22200u

J212V Output

12

+

C13

10u

+

C4470u

R9

33k

J3Input signal

12

J4Output signal

12

R1

220R

+

C5470u

R105k

R2

470R

+

C8100u

R810k

R6

1k

U1

78L05

V_in3

GN

D2

V_out1

R747k

+

C9

1u

Figure 2.2: Schematic diagram of the designed preamplifier

Figure 2.3: Designed preamplifier assembled on prototyping shield

8

..................................... 2.4. Preamplifier

Name Description

U1 5 V positive voltage regulator L78L05 (see [10])U2 Operational amplifier OP07C (see [11])R1 Resistor 220 ΩR2 Resistor 470 Ω

R3, R8 Resistor 10 kΩR4, R5, R7 Resistor 47 kΩ

R6 Resistor 1 kΩR9 Resistor 33 kΩR10 Trimmer resistor (potentiometer) 5 kΩ

C1, C3, C6, C7, C11 Ceramic capacitor 100 nFC2 Electrolytic capacitor 2200 µF, 16 V

C4, C5 Electrolytic capacitor 470 µF, 16 VC8 Electrolytic capacitor 100 µF, 6 V

C9, C10 Electrolytic capacitor 1 µF, 6 VC12, C13 Electrolytic capacitor 10 µF, 16 V

J1 Connector PC-GK2.1, connected to 12 V, 2 A DCpower supply

J2 Jumper providing 12 V power supplyJ3 Jack 3.5 mm mono connector for input signalJ4 Output signal jumper

Table 2.2: List of components used to assemble the preamplifier

The preamplifier is based on OP07C, a high precision op-amp (operationalamplifier) with very low input offset voltage. The whole circuit is to bepowered by single 12 V source connected to socket J1. The op-amp is wiredas a non-inverting amplifier and its non-inverting input is biased towardone-half the source voltage by voltage divider formed of resistors R4 andR5. Capacitor C9 filters out the DC component of the input signal and theAC component is then superposed on the bias voltage of 6 V. As a result,the amplified op-amp output signal is also superposed on this bias voltage.Capacitor C12 passes only the AC component. That together with presenceof the pull-down resistor R8 yield the output signal on jumper J4 without theDC component. In other words, the output signal is a waveform oscillatingaround 0 V.

Potentiometer R10 allows adjusting the gain. The minimum gain is

Amin = 1 + R9R10 = 7.6 (2.5)

and by adjusting the potentiometer we can set the gain arbitrarily high.However, the output signal is constrained by the supply voltage.

9

2. Construction of Rijke’s Tube ..............................Resistors R6 and R7 together with capacitor C10 filter out the power

supply noise by bypassing it to ground without affecting the input signal. Tosuppress the power supply noise even more, two RC circuits are used. Oneconsisting of R1, C3, and C4. The other consisting of R2, C5, and C6. C1,C2, C7, C8, and C11 are bypassing or decoupling capacitors.

Voltage regulator U1 serves to provide 5 V power supply for the microphone.In-series resistor R3 is necessary to avoid damaging the internal FET amplifier.

2.5 Speaker and Amplifier

At bottom end of the tube, universal loudspeaker Monacor SP-10/4 wasplaced. Its parameters are in tab. 2.3. It is driven by audio power amplifierTDA2030A (see [9]) that was bought as off-the-shelf module. Its maximumoutput power is 18 W.

Parameter Value

Impedance 4 ΩWattage 15–25 W

Dimensions 105 mm × 55 mm

Table 2.3: Loudspeaker Monacor SP-10/4 parameters

2.6 Data Acquisition

To acquire data from the microphone and actuate the loudspeaker, a computerwith data acquisition board MF624 by Hummusoft was used. It contains8 channel fast 14 bit ADC (analog-to-digital converter) and 8 channel 14bit DAC (Digital-to-analog converter.) The output of microphone preampli-fier was connected to the board’s ADC and the power audio amplifier wasconnected to board’s DAC.

The board was used together with Simulink Real-Time Windows Target.The reading frequency was 10 kHz. Fig. 2.4 and 2.5 depict Simulink modelsthat implement data collection and feedback loop control. To filter out high-frequency noise, low-pass filter with time constant τ = 1 ms was used in the

10

................................... 2.6. Data Acquisition

data acquisition. However, to eliminate time delay, it was not used in thefeedback loop.

Copyright1994-2016TheMathWorks,Inc.

Scope1

InfoInfo SwitchToSwitchToNormalModeNormalMode

AnalogInput

AnalogInputHumusoft

MF624[auto]

SpectrumAnalyzer

1

0.001s+1

Low-passFilter Scope2

Figure 2.4: Simulink model implementing data acquisition

Copyright1994-2016TheMathWorks,Inc.

Scope1

InfoInfo SwitchToSwitchToNormalModeNormalMode

AnalogOutput

AnalogOutputHumusoft

MF624[auto]

AnalogInput

AnalogInputHumusoft

MF624[auto]

SpectrumAnalyzer

Scope2

1

0.001s+1

Low-passFilter1

RepeatingSequenceStair

var

FromWorkspace

Scope4

K

Gain

Figure 2.5: Simulink model implementing feedback loop

11

12

Chapter 3

Experimental Evaluation

This chapter gives an evaluation of performed experiments. Firstly, the insta-bility of the system is observed. Secondly, feedback is introduced to stabilizethe system. Then, system identification is done. And finally, temperature ofthe heating coil is estimated.

3.1 Instability Observation

Fig. 3.1 shows the initial growth of oscillations of acoustic pressure in thetube. A close look at the oscillations is given in fig. 3.2 from which we can seedominant frequency of 131.6 Hz. This value is consistent with the expectationof standing half-wave in the tube and it satisfies (1.1). This frequency can alsobe seen in amplitude spectrum 3.3 of measured signal (for x = L/4) togetherwith the second harmonic of twice the frequency. As shown in spectrum 3.4,for heating coil position x = L/3 the second harmonic is not that significantas for x = L/4.

13

3. Experimental Evaluation................................

Time [s]0 2 4 6 8 10 12 14 16 18

Sou

nd

leve

l[V

]

-1.5

-1

-0.5

0

0.5

1

1.5

Initial Growth of Oscillations

Measured signal

Figure 3.1: Initial growth of oscillations

Time [s]20 20.01 20.02 20.03 20.04 20.05 20.06 20.07 20.08 20.09 20.1

Sou

nd

leve

l[V

]

-1.5

-1

-0.5

0

0.5

1

1.5

Close Look at the Oscillations

Measured signal

Figure 3.2: Close look at the oscillations

Frequency [Hz]0 50 100 150 200 250 300 350 400 450 500

Magnitude[V]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Amplitude Spectrum of Measured Signal (x = L=4)

Figure 3.3: Single-sided amplitude spectrum of measured signal for x = L/4

14

...................................3.2. Feedback Control

Frequency [Hz]0 50 100 150 200 250 300 350 400 450 500

Mag

nit

ude

[V]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Amplitude Spectrum of Measured Signal (x = L=3)

Figure 3.4: Single-sided amplitude spectrum of measured signal for x = L/3

3.2 Feedback Control

To stabilize the system, simple proportional regulator is sufficient for heatingcoil position x = L/3. For x = L/4, it turned out to be impossible. Values ofcontrol constant K around 0.3 stabilize the system making the humming goaway. The result of regulation is shown in fig. 3.5 where K changes from 0to 0.3. Since the the humming disappears and the regulation loop consists ofproportional feedback only, resulting signal fed to the speaker is zero. Thismeans that the regulator actually stabilizes the system. Hence, the processof stabilization should not be confused with active noise cancelation.

If the constant K is increased a little, the system becomes unstable again(fig. 3.6). This time a higher harmonic is dominant, specifically the 13th (fig.3.7).

15

3. Experimental Evaluation................................

Time [s]0 5 10 15 20 25

Sou

ndlevel[V

]

-4

-2

0

2

4

Feedback Stabilization

Measured signal

Time [s]0 5 10 15 20 25

Value[]

0

0.2

0.4

0.6

Control constant K

Figure 3.5: Stabilization of the system

Time [s]27 27.5 28 28.5 29 29.5 30

Sou

nd

leve

l[V

]

-0.5

0

0.5

Higher Harmonic Instability

Measured signal

Time [s]27 27.5 28 28.5 29 29.5 30

Value[]

0

0.2

0.4

0.6

Control constant K

Figure 3.6: Higher harmonic instability

16

................................. 3.3. System Identification

Frequency [Hz]0 500 1000 1500 2000 2500 3000

Mag

nit

ude

[V]

0

0.5

1

1.5

2

2.5

3

Amplitude Spectrum of Higher Harmonic Instability

Figure 3.7: Single-sided amplitude spectrum of higher harmonic instability

3.3 System Identification

The stable closed-loop system was identified by adding external signal w intothe regulation loop. For this signal, PRBS (pseudorandom binary sequence) ofduration 20 s was generated. The first ten seconds were used for identificationand the other ten were used for verification of identified system. In order tobe able to identify it, signal measured by the microphone and signal fed tothe loudspeaker were collected. From these, the system was identified usingparametric method known as ARX. The number of poles was set to 44 andthe number of zeros was set to 22. This method gave relatively accurate (seefig. 3.8 and 3.9). Fig. 3.10 frequency response obtained from ARX method.

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sou

nd

leve

l[V

]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Identi-ed and Original Systems Comparison

Identi-ed system responseOriginal system response

Figure 3.8: Comparision of responses to test signal

17

3. Experimental Evaluation................................

Time [s]0.1 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 0.11

Sou

nd

leve

l[V

]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Identi-ed and Original Systems Comparison

Identi-ed system responseOriginal system response

Figure 3.9: Comparision of responses to test signal (close look)

-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

From: u1 To: y1

10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 60

720

1440

2160

2880

3600

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

Figure 3.10: Frequency response from results of ARX

3.4 Heating Coil Temperature

The heating coil temperature can be estimated by measuring increase of itsresistance. Using approximation of linear temperature coefficient of resistancefor kanthal wire (obtained from [4])

α ≈ 2 × 10−5 K−1 (3.1)

we can estimate the wire temperature T as

T = T0 + R−R0αR0

(3.2)

18

............................... 3.4. Heating Coil Temperature

where R0 is the wire resistance at room temperature T0 and R is its resis-tance at temperature T . Resistance R0 is given by (2.3). Resistance R wasdetermined from measured voltage of the power supply

U = 48.06 V (3.3)

and measured current running through the wire when heated up

I = 7.77 A. (3.4)

Applying Ohm’s law we get

R = U

I= 6.19 Ω. (3.5)

Finally, by considering room temperature

T0 ≈ 20 C (3.6)

we obtain

T = 803 C. (3.7)

Nonetheless, this quantification is a very rough approximation only.

19

20

Chapter 4

Conclusion

In this thesis, the process of construction of Rijke’s tube is described. It alsodocumesnts a few experiments performed to illustrate basic properties of thetube. For instance, observations has shown that fundamental frequency of thetube dominates the spectrum of the oscillations which corresponds to standinghalf-wave inside the tube. Also, stabilization of the tube using proportionalregulator was achieved. Finally the tube as a system was identified usingparametrical methods.

All the performed experiments might be improved if frequency response ofused microphone and preamplifier was available. Knowledge of the responsewould make it possible to determine the signal distortion.

Possible direction for future research is measuring the sound at differentplaces and measuring the air flow in the tube. Another interesting obser-vations might be done by changing the position of the heating coil and theloudspeaker.

21

22

Appendix A

Bibliography

[1] J. P. Epperlein, B. Bamieh, and K. J. Astrom. Thermoacoustics andthe rijke tube: Experiments, identification, and modeling. IEEE ControlSystems, 35(2):57–77, April 2015.

[2] Jonathan Peter Epperlein. Topics in Modeling and Control of SpatiallyDistributed Systems. PhD thesis, University of California, Santa Barbara,2014.

[3] F.W. Hughes. Op-amp handbook. Prentice-Hall, 1986.

[4] Kanthal R© – Part of Sandvik Group. Kanthal A-1, Resistance heatingwire and resistance wire, January 2017.

[5] Konstantin Matveev. Thermoacoustic Instabilities in the Rijke Tube:Experiments and Modeling. PhD thesis, California Institute of TechnologyPasadena, California, 2003.

[6] Mean Well Enterprises Co., Ltd. SPV-1500 series, 1500W Single OutputPower Supply, August 2009.

[7] Winston Pun. Measurements of Thermo-Acoustic Coupling. PhD thesis,California Institute of Technology Pasadena, California, 2001.

[8] P. L. Rijke. Lxxi. notice of a new method of causing a vibration of theair contained in a tube open at both ends. Philosophical Magazine Series4, 17(116):419–422, 1859.

[9] STMicroelectronics. TDA2030A, 18 W hi-fi amplifier and 35 W driver,July 2011. Rev. 2.

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A. Bibliography.....................................[10] STMicroelectronics. L78L, Positive voltage regulators, June 2016. Rev.

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[11] Texas Instruments. OP07x Precision Operational Amplifiers, November2014. Rev. G.

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Appendix B

Abbreviations

AC Alternating currentADC Analog-to-digital converterDAC Digital-to-analog converterDAQ Data acquisitionDC Direct currentFET Field-effect transistorOp-amp Operational amplifierPC Personal computerPRBS Pseudorandom binary sequence

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Appendix C

CD Contents

The enclosed CD contains the following files.

. Lukas_Cerny_BP_May_2017.pdf – this bachelor thesis in pdf format. Lukas_Cerny_BP_Assignment.pdf – bachelor project assignment.MicrophonePreamplifier.pdf – schematic of the designed preamplifier. libs/ – directory containing libraries required to use DAQ board. ReadData.slx – Simulink model implementing data acquisition. FeedbackLoop.slx – Simulink model implementing feedback loop. InitMF624.m – MATLAB script initializing DAQ board

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